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I would like to obtain a generalization for the following:

$$ \bigg( A + \sum_{i = 1}^m c_i c_i^T \bigg)^{-1} $$

where $A \in \mathbb{R}^{n \times n}$ is a (symmetric) positive definite matrix and $c_i \in \mathbb{R}^{n \times 1}$, which has only one non-zero entry.

I was thinking of applying the Sherman-Morrison rank one update formula sequentially to each rank one matrix to obtain a generalization, but I was not able to do so. Would someone be able to help with a generalization for this so that it looks somewhat similar to if one were to perform the rank one update with just $c_1 c_1^T$ - i.e., if possible - since this is what I would really like to get?

Thank you.

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    What goes wrong when you try to apply Sherman-Morrison sequentially? – 2017-02-17

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You may have success applying the Woodbury identity, setting $C = I_{m\times m}$ and $$ U = V^T = \pmatrix{c_1 & c_2 & \cdots & c_m} $$

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    thank you for your help - My main problem was here: http://math.stackexchange.com/questions/2149513/is-the-following-inequality-satisfied I would appreciate any advice. thank you. – 2017-02-18